MathDB

A sequence

P_1, \dots, P_n

of points in the plane (not necessarily diferent) is carioca if there exists a permutation

a_1, \dots, a_n

of the numbers

1, \dots, n

for which the segments

P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}

are all of the same length.
Determine the greatest number

k

such that for any sequence of

k

points in the plane,

2023-k

points can be added so that the sequence of

2023

points is carioca.

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